184 3 Real Analysis
530.Letf, g:R^3 →Rbe twice continuously differentiable functions that are constant
along the lines that pass through the origin. Prove that on the unit ballB =
{(x,y,z)|x^2 +y^2 +z^2 ≤ 1 },
∫∫∫
B
f∇^2 gdV=
∫∫∫
B
g∇^2 fdV.
Here∇^2 =∂
2
∂x^2 +
∂^2
∂y^2 +
∂^2
∂z^2 is the Laplacian.
531.Prove Gauss’ law, which states that the total flux of the gravitational field through
a closed surface equals− 4 πGtimes the mass enclosed by the surface, whereGis
the constant of gravitation. The mathematical formulation of the law is
∫∫
S
−→
F ·−→ndS=− 4 πMG.
532.Let
−→
G(x, y)=
(
−y
x^2 + 4 y^2
,
x
x^2 + 4 y^2
, 0
)
.
Prove or disprove that there is a vector field
−→
F :R^3 →R^3 ,
−→
F (x, y, z)=(M(x, y, z), N (x, y, z), P (x, y, z)),
with the following properties:
(i)M, N, Phave continuous partial derivatives for all(x,y,z) =( 0 , 0 , 0 );
(ii) curl
−→
F =
−→
0 , for all(x,y,z) =( 0 , 0 , 0 );
(iii)
−→
F (x, y, 0 )=
−→
G(x, y).
533.Let
−→
F :R^2 →R^2 ,
−→
F (x, y)=(F 1 (x, y), F 2 (x, y))be a vector field, and let
G:R^3 →Rbe a smooth function whose first two variables arexandy, and the
third ist, the time. Assume that for any rectangular surfaceDbounded by the
curveC,
d
dt
∫∫
D
G(x, y, t)dxdy=−
∮
C
−→
F ·d
−→
R.
Prove that
∂G
∂t
+
∂F 2
∂x
+
∂F 1
∂y